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Properties

Label 3751.2.a.m
Level $3751$
Weight $2$
Character orbit 3751.a
Self dual yes
Analytic conductor $29.952$
Analytic rank $0$
Dimension $15$
CM no
Inner twists $1$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3751,2,Mod(1,3751)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3751, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3751.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 3751 = 11^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3751.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [15,1,2,17,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(29.9518857982\)
Analytic rank: \(0\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{15} - x^{14} - 23 x^{13} + 21 x^{12} + 204 x^{11} - 160 x^{10} - 880 x^{9} + 535 x^{8} + 1918 x^{7} + \cdots + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 23 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{14}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{12} q^{3} + (\beta_{2} + 1) q^{4} + (\beta_{14} + 1) q^{5} + (\beta_{14} - \beta_{11} + \cdots + \beta_1) q^{6} + \beta_{8} q^{7} + (\beta_{3} + \beta_1) q^{8} + (\beta_{13} - \beta_{12} - \beta_{8} + \cdots + 1) q^{9}+ \cdots + ( - 3 \beta_{14} + 2 \beta_{13} + \cdots - 3) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q + q^{2} + 2 q^{3} + 17 q^{4} + 8 q^{5} - 2 q^{6} + 3 q^{8} + 25 q^{9} - 15 q^{10} + 11 q^{12} - 4 q^{13} + 9 q^{14} + 15 q^{15} + 29 q^{16} - 2 q^{17} + 4 q^{18} + 5 q^{19} + 17 q^{20} - 15 q^{21}+ \cdots - 31 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{15} - x^{14} - 23 x^{13} + 21 x^{12} + 204 x^{11} - 160 x^{10} - 880 x^{9} + 535 x^{8} + 1918 x^{7} + \cdots + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 5\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 397 \nu^{14} - 379 \nu^{13} - 8937 \nu^{12} + 7879 \nu^{11} + 76554 \nu^{10} - 58972 \nu^{9} + \cdots - 1954 ) / 1048 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 295 \nu^{14} - 343 \nu^{13} - 6691 \nu^{12} + 7329 \nu^{11} + 58118 \nu^{10} - 57494 \nu^{9} + \cdots - 5444 ) / 786 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 925 \nu^{14} + 1675 \nu^{13} + 20101 \nu^{12} - 36063 \nu^{11} - 163850 \nu^{10} + 287720 \nu^{9} + \cdots - 14170 ) / 1572 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 635 \nu^{14} + 987 \nu^{13} + 14003 \nu^{12} - 21127 \nu^{11} - 116864 \nu^{10} + 166784 \nu^{9} + \cdots - 3744 ) / 786 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 2987 \nu^{14} - 5069 \nu^{13} - 65639 \nu^{12} + 109065 \nu^{11} + 543718 \nu^{10} - 868780 \nu^{9} + \cdots + 33530 ) / 3144 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 3485 \nu^{14} + 5291 \nu^{13} + 76913 \nu^{12} - 113199 \nu^{11} - 642298 \nu^{10} + 892348 \nu^{9} + \cdots - 18590 ) / 3144 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 1761 \nu^{14} + 2679 \nu^{13} + 38869 \nu^{12} - 57195 \nu^{11} - 324426 \nu^{10} + 449780 \nu^{9} + \cdots - 3238 ) / 1048 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 6523 \nu^{14} + 10509 \nu^{13} + 144199 \nu^{12} - 224897 \nu^{11} - 1206238 \nu^{10} + \cdots - 13386 ) / 3144 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 3837 \nu^{14} - 6155 \nu^{13} - 84705 \nu^{12} + 131639 \nu^{11} + 706958 \nu^{10} - 1038908 \nu^{9} + \cdots + 22178 ) / 1572 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 4135 \nu^{14} - 6553 \nu^{13} - 91123 \nu^{12} + 140325 \nu^{11} + 758498 \nu^{10} - 1109060 \nu^{9} + \cdots + 29482 ) / 1572 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 8917 \nu^{14} + 14051 \nu^{13} + 196729 \nu^{12} - 300655 \nu^{11} - 1640298 \nu^{10} + \cdots - 66278 ) / 3144 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{13} + \beta_{10} - \beta_{8} + 7\beta_{2} + 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{14} - \beta_{13} + \beta_{12} - \beta_{9} + \beta_{8} - \beta_{6} + \beta_{4} + 9\beta_{3} - \beta_{2} + 29\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2 \beta_{14} + 11 \beta_{13} - \beta_{11} + 10 \beta_{10} - 10 \beta_{8} - \beta_{6} + \beta_{4} + \cdots + 97 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 12 \beta_{14} - 10 \beta_{13} + 11 \beta_{12} - \beta_{11} + \beta_{10} - 11 \beta_{9} + 9 \beta_{8} + \cdots - 17 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 28 \beta_{14} + 91 \beta_{13} + 2 \beta_{12} - 13 \beta_{11} + 81 \beta_{10} - \beta_{9} - 77 \beta_{8} + \cdots + 618 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 105 \beta_{14} - 78 \beta_{13} + 89 \beta_{12} - 18 \beta_{11} + 17 \beta_{10} - 92 \beta_{9} + \cdots - 194 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 277 \beta_{14} + 685 \beta_{13} + 32 \beta_{12} - 122 \beta_{11} + 611 \beta_{10} - 14 \beta_{9} + \cdots + 4040 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 812 \beta_{14} - 568 \beta_{13} + 638 \beta_{12} - 213 \beta_{11} + 190 \beta_{10} - 699 \beta_{9} + \cdots - 1859 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 2388 \beta_{14} + 4952 \beta_{13} + 343 \beta_{12} - 1006 \beta_{11} + 4449 \beta_{10} - 131 \beta_{9} + \cdots + 26835 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 5881 \beta_{14} - 4058 \beta_{13} + 4282 \beta_{12} - 2093 \beta_{11} + 1764 \beta_{10} - 5083 \beta_{9} + \cdots - 16187 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 19180 \beta_{14} + 35109 \beta_{13} + 3109 \beta_{12} - 7739 \beta_{11} + 31720 \beta_{10} + \cdots + 180204 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−2.66811
−2.60220
−1.85880
−1.74615
−0.930390
−0.594714
−0.584236
0.0599692
0.240763
1.05850
1.15277
2.14697
2.31885
2.42107
2.58571
−2.66811 1.95656 5.11884 3.96532 −5.22034 3.16640 −8.32141 0.828144 −10.5799
1.2 −2.60220 −0.797979 4.77145 0.0550764 2.07650 −1.92797 −7.21187 −2.36323 −0.143320
1.3 −1.85880 2.89552 1.45512 1.82539 −5.38218 −4.44083 1.01282 5.38403 −3.39303
1.4 −1.74615 −1.55132 1.04905 1.03434 2.70884 5.16265 1.66050 −0.593409 −1.80611
1.5 −0.930390 −1.59384 −1.13438 −0.0348804 1.48289 −2.18154 2.91619 −0.459666 0.0324524
1.6 −0.594714 −3.16059 −1.64631 3.53087 1.87965 −2.16092 2.16852 6.98930 −2.09986
1.7 −0.584236 3.13214 −1.65867 −3.75699 −1.82991 0.671500 2.13753 6.81029 2.19497
1.8 0.0599692 −2.04429 −1.99640 −1.13818 −0.122595 1.94033 −0.239661 1.17913 −0.0682558
1.9 0.240763 0.685295 −1.94203 −1.44419 0.164993 −4.65408 −0.949094 −2.53037 −0.347707
1.10 1.05850 3.28286 −0.879570 4.02132 3.47492 1.88072 −3.04804 7.77719 4.25658
1.11 1.15277 0.107389 −0.671122 3.27597 0.123795 −2.63305 −3.07919 −2.98847 3.77644
1.12 2.14697 −0.419420 2.60949 −3.42401 −0.900482 −1.39219 1.30855 −2.82409 −7.35126
1.13 2.31885 −3.27716 3.37707 −2.64797 −7.59926 3.57154 3.19322 7.73981 −6.14026
1.14 2.42107 0.349901 3.86158 2.49150 0.847135 4.40746 4.50700 −2.87757 6.03208
1.15 2.58571 2.43493 4.68589 0.246438 6.29603 −1.41001 6.94494 2.92891 0.637217
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.15
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(11\) \( -1 \)
\(31\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3751.2.a.m yes 15
11.b odd 2 1 3751.2.a.l ✓ 15
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3751.2.a.l ✓ 15 11.b odd 2 1
3751.2.a.m yes 15 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3751))\):

\( T_{2}^{15} - T_{2}^{14} - 23 T_{2}^{13} + 21 T_{2}^{12} + 204 T_{2}^{11} - 160 T_{2}^{10} - 880 T_{2}^{9} + \cdots + 4 \) Copy content Toggle raw display
\( T_{3}^{15} - 2 T_{3}^{14} - 33 T_{3}^{13} + 59 T_{3}^{12} + 418 T_{3}^{11} - 621 T_{3}^{10} - 2590 T_{3}^{9} + \cdots + 64 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{15} - T^{14} + \cdots + 4 \) Copy content Toggle raw display
$3$ \( T^{15} - 2 T^{14} + \cdots + 64 \) Copy content Toggle raw display
$5$ \( T^{15} - 8 T^{14} + \cdots - 23 \) Copy content Toggle raw display
$7$ \( T^{15} - 71 T^{13} + \cdots - 612223 \) Copy content Toggle raw display
$11$ \( T^{15} \) Copy content Toggle raw display
$13$ \( T^{15} + 4 T^{14} + \cdots + 53824 \) Copy content Toggle raw display
$17$ \( T^{15} + 2 T^{14} + \cdots + 83116480 \) Copy content Toggle raw display
$19$ \( T^{15} + \cdots - 123000207 \) Copy content Toggle raw display
$23$ \( T^{15} - 17 T^{14} + \cdots - 33856 \) Copy content Toggle raw display
$29$ \( T^{15} - 7 T^{14} + \cdots + 304704 \) Copy content Toggle raw display
$31$ \( (T + 1)^{15} \) Copy content Toggle raw display
$37$ \( T^{15} - 17 T^{14} + \cdots + 2324544 \) Copy content Toggle raw display
$41$ \( T^{15} + 7 T^{14} + \cdots + 69046780 \) Copy content Toggle raw display
$43$ \( T^{15} + \cdots + 166767737280 \) Copy content Toggle raw display
$47$ \( T^{15} + \cdots - 6733931579 \) Copy content Toggle raw display
$53$ \( T^{15} + \cdots - 5191182144 \) Copy content Toggle raw display
$59$ \( T^{15} + \cdots - 6410448799 \) Copy content Toggle raw display
$61$ \( T^{15} + \cdots + 9039259392 \) Copy content Toggle raw display
$67$ \( T^{15} + \cdots + 422827644809 \) Copy content Toggle raw display
$71$ \( T^{15} - 35 T^{14} + \cdots + 610173 \) Copy content Toggle raw display
$73$ \( T^{15} + \cdots - 3007064896 \) Copy content Toggle raw display
$79$ \( T^{15} + \cdots - 9362484249920 \) Copy content Toggle raw display
$83$ \( T^{15} + \cdots + 381563035968 \) Copy content Toggle raw display
$89$ \( T^{15} + \cdots - 44638289226816 \) Copy content Toggle raw display
$97$ \( T^{15} + \cdots + 670690067951 \) Copy content Toggle raw display
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